Division is one operation that reverses the result of multiplication. However, division has limitations: it's impossible to divide by 0. Other than that, if a = b then a / r = b / r. However, concealed division by 0 results in errors and what's commonly known as arithmetic paradoxes. Here is a well known one: assume a = b and that both are positive numbers. Write

If two numbers are equal, their squares are also equal. Is the reverse true? That is, is it true that squares of two numbers are equal provided the numbers are equal, to start with? In short, does u = v imply √u = √v? Or simpler yet: is the square root of a number unique?

Assume the answer is unqualified "yes" and let's see where this will lead us.

We assumed at the outset that a = b. But then, on step #6, divided the equation by b - a which is, by our assumption, zero. Dividing by 0 makes no sense and may lead to unpredictable results.

The problem here is with a careless use of the term "square root" and the corresponding symbol √ . Squaring is a many-to-one operation: 2² = (-2)², implying that the converse operation is not defined uniquely. If a non-zero real number has a real square root, it has two square roots, i.e., there are two numbers whose square equals the given one. For a given R, one of such numbers is denoted √R while the other -√R. In short, ±√R. One of this is positive, the other negative. As a matter of definition, √R stands for the positive number, e.g., √25 = 5; writing or assuming √25 = -5 is a common mistake.

The above derivation played out on that difference between the uniqueness of √ and a lack of uniqueness in interpreting what stands behind the verbal "square root". The opening questions were designed to conceal this difference and hopefully confuse the reader.

If two numbers are equal, their squares are also equal. Is the reverse true? That is, is it true that square roots of two numbers are equal provided the numbers are equal, to start with? (Innocuously this is where the confusion starts: the square root is not unique. So, which one is the question about?) In short, does u = v imply √u = √v? (The answer to this question is an unqualified "Yes". By the definition of √ this is true. This question and the implied answer serve as a diversion that sets a trap for the next question.) Or simpler yet: is the square root of a number unique? (After the previous question, there may be an impulse to answer "Yes" to that question too. We go on and make that assumption.) Assume the answer is unqualified "yes" and let's see where this will lead us.

The correct result on step 7 would be (n + 1) - (2n + 1)/2 = -(n - (2n + 1)/2). Ignoring the fact that the number on the right is negative (and, therefore, cannot serve as √ of another number) leads to a contradiction.